Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2009-10-04
Journal of Mathematical Physics 51, 022108 (2010) (14 pages)
Physics
High Energy Physics
High Energy Physics - Theory
20 pages; minor corrections were made; new Appendix was added
Scientific paper
We study a quantum model with non-isotropic two-dimensional oscillator potential but with additional quadratic interaction $x_1x_2$ with imaginary coupling constant. It is shown, that for a specific connection between coupling constant and oscillator frequences, the model {\it is not} amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of $R-$separation of variables in two-dimensional systems is discussed.
Cannata Francesco
Ioffe Mikhail V.
Nishnianidze D. N.
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